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Nonextensive Perfect Hydrodynamics – a Model of Dissipative Relativistic Hydrodynamics?

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Nonextensive Perfect Hydrodynamics – a Model of Dissipative Relativistic Hydrodynamics? Cent. Eur. J. Phys. • 7(3) • 2009 • 432-443 DOI: 10.2478/s11534-008-0163-5 Central European Journal of Physics Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics? Research Article Takeshi Osada1∗, Grzegorz Wilk2† 1 Theoretical Physics Lab., Faculty of Knowledge Engineering, Musashi Institute of Technology, Setagaya-ku, Tokyo 158-8557, Japan 2 The Andrzej Sołtan Institute for Nuclear Studies, Theoretical Physics Department, ul. Hoża 69, 00-681 Warsaw, Poland Received 14 October 2008; accepted 19 December 2008 Abstract: We demonstrate that nonextensive perfect relativistic hydrodynamics (q-hydrodynamics) can serve as a model of the usual relativistic dissipative hydrodynamics (d-hydrodynamics) therefore facilitating consider- ably its applications. As an illustration, we show how using q-hydrodynamics one gets the q-dependent expressions for the dissipative entropy current and the corresponding ratios of the bulk and shear viscosities to entropy density, ζ/s and η/s respectively. PACS (2008): 05.70.Ln, 12.38.Mh, 12.40.Ee Keywords: nonextensive statistical mechanics • hydrodynamic models • high energy collisions • multiparticle produc- tion processes © Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction be very effectively described by using a thermodynamic approach in conjunction with hydrodynamical flow. In this so called Landau Hydrodynamic Model, the secondaries were considered to be a product of the decay of some Hydrodynamics is well established as an effective ap- kind of hadronic fluid, produced in such collisions, which proach to flow phenomena. Here we shall present its was expanding before hadronization [1–3]. This model has nonextensive relativistic version from the point of view of recently been updated to describe recent experimental high energy collision physics [1–14]. The characteristic data [4, 5]. Since then there have been a number of feature of such a processes is the production of a large successful attempts to develop new solutions for both the number of secondaries (multiplicities at present approach Landau model [6] and for the so called Hwa-Bjorken ver- ∼ 103). Already in 1953, when there were only ∼ 10 sion of the hydrodynamical model [7, 8], for which a new particles produced and registered, mostly in cosmic rays class of solutions has been found [9, 10]. Hydrodynamic experiments, it was found [1–3] that such processes can models of different types have therefore been frequently used in the phenomenological description of multiparticle ∗ production processes at high energies, especially for high E-mail: [email protected] † E-mail: [email protected] (Corresponding author) energy nuclear collisions [11–14]. These are of special 432 Takeshi Osada, Grzegorz Wilk interest to us (being currently investigated at RHIC at fully clear (cf., [48])1. Brookhaven, with the newly commissioned LHC at CERN joining soon) because it is widely believed that in such collisions a new form of hadronic matter, the so called The physical picture emerging from the above experience Quark-Gluon-Plasma (QGP), will be produced [15–18]. is that, instead of a strict local thermal equilibrium customarily assumed in all applications of statistical models (including hydrodynamic), one rather encounters a kind of stationary state, which already includes some So far all hydrodynamic models in this field have been interactions. It can be introduced in different ways. based on the usual Boltzmann-Gibbs (BG) form of For example, in [54] it was a random distortion of the statistical mechanics. The only works discussing some energy and momentum conservation caused by the general features of the nonextensive hydrodynamics, surrounding system. This results in the emergence of which we are aware of [19–21], use a nonrelativistic ap- some nonextensive equilibrium. In [55–59] the two-body proach and are therefore not suitable for the applications energy composition is replaced by some generalized h E ;E we are interested in. On the other hand it is known energy sum ( 1 2), which is assumed to be associative that an approach based on non-extensive statistical but which is not necessarily simple addition and contains mechanics (used mainly in the form proposed by Tsallis contributions stemming from pair interaction (in the [22–28] with only one new parameter, the nonextensivity simplest case). It turns out that, under quite general parameter q) describes different sets of data in a better assumptions about the function h, the division of the way than the usual statistical models based on BG total energy among free particles can be done. Different statistics, cf., [22–28] for general examples and [29–47] forms of the function h then lead to different forms of the for applications to multi-particle production processes. entropy formula, among which one encounters the known Roughly speaking, all observed effects amount to a Tsallis entropy. The origin of this kind of thinking can broadening of the respective spectra of the observed be traced back to the analysis of the q-Hagedorn model secondaries (both in transverse momentum space and proposed some time ago in [60]. in rapidity space), they take the form of q-exponents instead of the naively expected usual exponents: 1/(1−q) exp(−X/T ) =⇒ expq(−X/T ) = [1 − (1 − q)X/T ] . All phenomenological applications of hydrodynamic From these studies emerged a commonly accepted models to the recent multiparticle production processes interpretation of the nonextensivity parameter q (in fact, show that, although perfect (nonviscous) hydrodynamics |q − 1|), as the measure of some intrinsic fluctuations successfully describes most RHIC data [6, 61], there are characteristic of the hadronizing systems under consid- indications that a hadronic fluid cannot be totally ideal. eration [29–31]. For q > 1, in transverse momentum For example, the perfect fluid dynamical calculations with space, q could represent fluctuation of the temperature, color glass condensate initial states could not reproduce T , corresponding to some specific heat parameter C. the elliptic flow data [63], indicating a necessity to use In this case q − 1 = C and therefore it should be some kind of viscous fluid description. A nonrelativistic inversely proportional to the volume of the interaction viscous fluid is usually described by the first order region. This effect is indeed observed [46, 47]. In rapidity space these are fluctuations of the so called 1 One should keep in mind, however, that there are du- partition temperature T E/hni , pt = [42–45], which are alities in the non-extensive approach, i.e., that both q precisely the same fluctuations that lead to the Negative and 1/q can be used as the nonextensivity parameter de- Binomial form of the observed multiplicity distributions, pending on the normalization of original or q-powered P(n; hni; k), with its characteristic parameter k given by probabilities. Also, when considering the particle-hole k = 1/(q − 1) [42–45, 48]. In the case of q < 1, the symmetry in the q-Fermi distribution, f(−E; T ; µ; q) = − f E;T; −µ; − q , in a plasma containing both par- interpretation is not at present clear [31]. It seems that 1 ( 2 ) ticles and antiparticles both q and − q occur (µ de- the first role of the parameter q is to restrict the allowed 2 notes the chemical potential here). These dual possibili- phase space [49]. Actually, the conjecture associating ties warn us that a theory requiring that only q > has q 1 with fluctuations has already been formalized as a physical meaning is still incomplete [22–28]. These points superstatistics new branch of statistical mechanics called deserve further considerations which are, however, going [50, 51]. It should be noted that there are also arguments outside the scope of this paper. On the other hand, in the connecting nonextensivity with some special dynamical present paper we are not considering plasmas containing correlations existing in the system under consideration both particles and antiparticles but only massive pions [52, 53], but their connection with fluctuations is not yet which are assumed to obey the q-Boltzmann distribution. 433 Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics? Navier-Stokes equations. However, one needs their esis is now assumed in nonextensive form: special relativistic generalization and this turns out to be acausal and unstable [64, 65] (see also [66–71]). h f ; f f f ; q[ q q1] = expq [lnq q + lnq q1] One therefore looks towards the extended, second order theories accepting all problems connected with their for- where 1 1−q (1) mulation and practical applications [72–87]. Physically, expq(X) = [1 + (1 − q)X] ; the difference is that first order theories are based on (1−q) X − 1 the local equilibrium hypothesis, in which independent lnq(X) = : 1 − q variables are used, whereas in higher order theories only the fluxes of the local equilibrium theory appear as f x; p q independent variables. In particular, the entropy vector Here q( ) is the -version of the phase space distri- h f ; f q is quadratic in the fluxes, containing terms characterizing bution function, whereas q[ q q1] is the -version of the the deviation from local equilibrium. This situation, correlation function related to the presence of two par- x plus our experience with the nonextensive formalism, ticles in the same space-time position but with differ- p p [29–37, 42–48] prompted us to investigate the simple ent four-momenta and 1, respectively. By postulat- assuming nonextensive formulation of the perfect hydrodynamic ing Eq. (1) we are, in fact, that, instead of a a kind of stationary state is be- model, a perfect q-hydrodynamics [88–90]. It turned out strict (local) equilibrium, ing formed that this describes the experimental data fairly well. In , which already includes some interactions and q addition, an apparently unexpected feature appeared, which is summarily characterized by a parameter ; very namely the possibility that (relatively simple and first much in the spirit of [54–60] mentioned before.
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